Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 39–46

CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY–LITTLEWOOD MAXIMAL OPERATOR

Joaquim Martín and Javier Soria

Universitat Autònoma de Barcelona, Department of Mathematics ES-08193 Bellaterra (Barcelona), Spain; jmartin@mat.uab.es University of Barcelona, Department of Applied Mathematics and Analysis ES-08071 Barcelona, Spain; soria@ub.edu

Abstract. We characterize the rearrangement invariant spaces for which there exists a nonconstant fixed point, for the Hardy–Littlewood maximal operator (the case for the spaces L^{p}(R^{n}) was first considered in [7]). The main result that we prove is that the space L^{n/(n−2),∞}(R^{n}) ∩ L^{∞}(R^{n}) is minimal among those having this property.

1. Introduction

The centered Hardy–Littlewood maximal operator ℳ is defined on the Lebesgue space L_{loc}^{1}(R^{n}) by

ℳ f(x) = sup_{r>0} \frac{1}{|B_{r}|} ∫_{B_{r}} |f(x − y)| dy,

where |B_{r}| denotes the measure of the Euclidean ball B_{r} centered at the origin of R^{n}.

In this paper we study the existence of non-constant fixed points of the maximal operator ℳ (i.e., ℳ f = f) in the framework of the rearrangement invariant (r.i.) functions spaces (see Section 2 below). We will use some of the estimates proved in [7], where the case L^{p}(R^{n}) was studied, and show that they can be sharpened to obtain all the rearrangement invariant norms with this property (in particular we extend Korry’s result to the end point case p = n/(n−2), where the weak-type spaces have to considered). The main argument behind this problem is the existence of a minimal space L^{n/(n−2),∞}(R^{n}) ∩ L^{∞}(R^{n}) contained in all the r.i. spaces with the fixed point property.

2000 Mathematics Subject Classification: Primary 42B25, 46E30.

The research of the first-named author was partially supported by Grants MTM2004-02299, 2001SGR00069 and by Programa Ramón y Cajal (MCYT) and that of the second author was partially supported by Grants MTM2004-02299, 2001SGR00069.

2. Background on rearrangement invariant spaces

Since we work in the context of rearrangement invariant spaces it will be convenient to start by reviewing some basic definitions about these spaces.

A rearrangement invariant space X = X(R^{n}) (r.i. space) is a Banach function space on R^{n} endowed with a norm ∥ · ∥_{X(R^{n})} such that

∥f∥_{X(R^{n})} = ∥g∥_{X(R^{n})}

whenever f^{∗} = g^{∗}. Here f^{∗} stands for the non-increasing rearrangement of f, i.e., the non-increasing, right-continuous function on [0, ∞) equimeasurable with f.

An r.i. space X(R^{n}) has a representation as a function space on \overline{X} (0, ∞) such that

∥f∥_{X(R^{n})} = ∥f^{∗}∥_{\overline{X} (0,∞)}.

Any r.i. space is characterized by its fundamental function

ϕ_{X}(s) = ∥χ_{E}∥_{X(R^{n})}

(here E is any subset of R^{n} with |E| = s) and the fundamental indices

β̄_{X} = inf_{s>1} \frac{log M_{X}(s)}{log s} and \underline{β}_{X} = sup_{s<1} \frac{log M_{X}(s)}{log s},

where

M_{X}(s) = sup_{t>0} \frac{ϕ_{X}(ts)}{ϕ_{X}(t)}, s > 0.

It is well known that

0 ≤ \underline{β}_{X} ≤ β̄_{X} ≤ 1.

(We refer the reader to [2] for further information about r.i. spaces.)

3. Main result

Before formulating our main result, it will be convenient to start with the following remarks (see [7]):

Remark 3.1. By Lebesgue’s differentiation theorem one easily obtains that

|f(x)| ≤ ℳ f(x) a.e. x ∈ R^{n};

thus f is a fixed point of ℳ , if and only if f is positive and

\frac{1}{|B(x, r)|} ∫_{B(x,r)} f(y) dy ≤ f(x) a.e. x ∈ R^{n},

or equivalently f is a positive super-harmonic function (i.e. △f ≤ 0, where △ is the Laplacian operator).

Remark 3.2. If f is a non-constant fixed point of ℳ , and φ ≥ 0 belongs to the Schwartz class 𝒮 (R^{n}), with ∫_{R^{n}} φ(x) dx = 1, then the function f_{t}(x) = (f ∗φ_{t})(x), with φ_{t}(x) = t^{−n}φ(x/t) is also a non-constant fixed point of ℳ which belongs to 𝒞^{∞}(R^{n}) (notice that using the Lebesgue differentiation theorem, there exists some t > 0 such that f_{t} is non-constant, since f is non-constant). In particular if X(R^{n}) is an r.i. space and f ∈ X(R^{n}) is a non-constant fixed point of ℳ , since 𝒮 (R^{n}) ⊂ L^{1}(R^{n}) ∩ L^{∞}(R^{n}) we get that f_{t} ∈ X(R^{n}) ∩ 𝒞^{∞}(R^{n}) is a non-constant fixed point of ℳ .

Remark 3.3. Using the theory of weighted inequalities for ℳ (see [5]), if ℳ f = f, in particular f ∈ A_{1} (the Muckenhoupt weight class), and hence f(x) dx defines a doubling measure. Hence, f ∉ L^{1}(R^{n}). Also, using the previous remark we see that if f ∈ L^{p}(R^{n}) is a fixed point, then f ∈ L^{q}(R^{n}), for all p ≤ q ≤ ∞.

Definition 3.4. Given an r.i. space X(R^{n}), we define

D_{I_{2}}(X(R^{n})) = {f ∈ L^{0}(R^{n}) : ∥I_{2}f∥_{X(R^{n})} < ∞},

where I_{2} is the Riesz potential,

(I_{2}f)(x) = ∫_{R^{n}} |x − y|^{2−n}f(y) dy.

It is not hard to see that the space D_{I_{2}}(X(R^{n})) is either trivial or is the largest r.i. space which is mapped by I_{2} into X(R^{n}), and is also related with the theory of the optimal Sobolev embeddings (see [4] and the references quoted therein).

Theorem 3.5. Let X(R^{n}) be an r.i. space. The following statements are equivalent:

(1) There is a non-constant fixed point f ∈ X(R^{n}) of ℳ .

(2) n ≥ 3 and |x|^{2−n}χ_{{x:|x|>1}}(x) ∈ X(R^{n}).

(3) n ≥ 3 and χ_{[0,1]}(t) + t^{2/n−1}χ_{[1,∞)}(t) ∈ \overline{X} (0, ∞).

(4) n ≥ 3 and (L^{n/(n−2),∞}(R^{n}) ∩ L^{∞}(R^{n})) ⊂ X(R^{n}).

(5) n ≥ 3 and D_{I_{2}}(X(R^{n})) ≠ {0}.

Proof. (1) ⇒ (2) Since if n = 1 or n = 2, the only positive super-harmonic functions are the constant functions (see [8, Remark 1, p. 210]), necessarily n ≥ 3. Moreover, it is proved in [7] that, if f ∈ 𝒞^{∞}(R^{n}) is a non-constant fixed point of ℳ , then

f(x) ≥ c|x|^{2−n}χ_{{x:|x|>1}}(x).

Since f ∈ X(R^{n}), then |x|^{2−n}χ_{{x:|x|>1}}(x) ∈ X(R^{n}).

(2) ⇒ (3) Since if |x|^{2−n}χ_{{x:|x|>1}}(x) ∈ X(R^{n}), then
F (x) = χ_{{x:|x|≤1}}(x) + |x|^{2−n}χ_{{x:|x|>1}}(x) ∈ X(R^{n}).

An easy computation shows that

F^{∗}(t) ≃ χ_{[0,1]}(t) + t^{2/n−1}χ_{[1,∞)}(t).
(3) ⇒ (4) Since f ∈ (L^{n/(n−2),∞}(R^{n}) ∩ L^{∞}(R^{n})) if and only if

sup_{t>0} f^{∗}(t)W (t) < ∞,

where W (t) = max(1, t^{1−2/n}), we have that

f^{∗}(t) ≤ ∥f∥_{L^{n/(n−2),∞}(R^{n})∩L^{∞}(R^{n})}W^{−1}(t)

and since W^{−1}(t) = χ_{[0,1]}(t) + t^{2/n−1}χ_{[1,∞)} ∈ \overline{X} (0, ∞) we have that

∥f∥_{X(R^{n})} = ∥f^{∗}∥_{\overline{X} (0,∞)} ≤ c∥f∥_{L^{n/(n−2),∞}(R^{n})∩L^{∞}(R^{n})}

with c = ∥W^{−1}∥_{\overline{X} (0,∞)} .

(4) ⇒ (5) Since (see [9] and [1])

(I_{2}f)^{∗}(t) ≤ c_{1}(t^{2/n−1} ∫_{0}^{t} f^{∗}(s) ds + ∫_{t}^{∞} f^{∗}(s)s^{2/n−1} ds) ≤ c_{2}(I_{2}f^{0})^{∗}(t)

where f^{0}(x) = f^{∗}(c_{n}|x|^{n}), c_{n} = measure of the unit ball in R^{n}. (Observe that (f^{0})^{∗} = f^{∗}). Rewriting the middle term in the above inequalities, using Fubini’s theorem, we get

(I_{2}f)^{∗}(t) ≤ d_{1}(\frac{n}{n − 2} ∫_{t}^{∞} f^{∗∗}(s)s^{2/n−1} ds) ≤ d_{2}(I_{2}f^{0})^{∗}(t),

where f^{∗∗}(t) = t^{−1} ∫_{0}^{t} f^{∗}(s) ds. Thus, f ∈ D_{I_{2}}(X(R^{n})) if and only if

(1)  ∥∫_{t}^{∞} f^{∗∗}(s)s^{2/n−1} ds∥_{\overline{X} (0,∞)} < ∞.

Since

F (t) = ∫_{t}^{∞} χ_{[0,1]}^{∗∗}(s)s^{2/n−1} ds = c(χ_{[0,1]}(t) + t^{2/n−1}χ_{[1,∞)}(t))

is a decreasing function, and

\begin{aligned}F^{0}(x) & = F (c_{n}|x|^{n}) \\ & ≃ (χ_{\{x:|x|≤1\}}(x) + |x|^{2−n}χ_{\{x:|x|>1\}}(x)) ∈ L^{n/(n−2),∞}(R^{n}) ∩ L^{∞}(R^{n}) \end{aligned}

we get that χ_{[0,1]}^{∘} ∈ D_{I_{2}}(X(R^{n})).

Another argument to prove this part is the following:

Since, if n ≥ 3 (see [2, Theorem 4.18, p. 228])

I_{2}: L^{1}(R^{n}) → L^{n/(n−2),∞}(R^{n}) and I_{2}: L^{n/2,1}(R^{n}) → L^{∞}(R^{n})

is bounded, we have that

I_{2}: (L^{1}(R^{n}) ∩ L^{n/2,1}(R^{n})) → (L^{n/(n−2),∞}(R^{n}) ∩ L^{∞}(R^{n})) ⊂ X(R^{n})

is bounded, and hence L^{1}(R^{n}) ∩ L^{n/2,1}(R^{n}) ⊂ D_{I_{2}}(X(R^{n})).

(5) ⇒ (1) Since n ≥ 3, we can use the classical formula of potential theory (see [10, p. 126])

−h = △(I_{2}h)

to conclude that there is a positive function f = I_{2}χ_{[0,1]}^{∘} ∈ X(R^{n}). Then 0 ≤ f_{t} = I_{2}(χ_{[0,1]}^{∘} ∗ φ_{t}) ∈ X(R^{n}) ∩ 𝒞^{∞}(R^{n}) and △f_{t} ≤ 0.

We now consider particular examples, like the Lorentz spaces:

Corollary 3.6. Let 1 ≤ p < ∞, and assume Λ^{p}(R^{n}, w) is a Banach space (i.e., w ∈ B_{p} if 1 < p < ∞ or p ∈ B_{1,∞} if p = 1, see [3]). Then, there exists a non-constant function f ∈ Λ^{p}(R^{n}, w) such that ℳ (f) = f if and only if n ≥ 3 and

∫_{1}^{∞} \frac{w(t)}{t^{p(1−2/n)}} dt < ∞.

In particular, this condition always holds, for p > 1 and n large enough.

Proof. The integrability condition follows by using the previous theorem. Now, if w ∈ B_{p}, then there exists an ε > 0 such that w ∈ B_{p−ε}, and hence it suffices to take n > 2/ε. Observe that if w = 1 and p = 1, then Λ^{1}(R^{n}, w) = L^{1}(R^{n}), which does not have the fixed point property for any dimension n.

Corollary 3.7. Let 1 ≤ p, q ≤ ∞ (if p = 1 we only consider q = 1). Then, there exists a non-constant function f ∈ L^{p,q}(R^{n}) such that ℳ (f) = f if and only if n ≥ 3 and

\{ \begin{matrix}n/(n − 2) < p ≤ ∞ or \\ p = n/(n − 2) and q = ∞. \end{matrix}

Corollary 3.8. (See [7]) Let 1 ≤ p ≤ ∞. There exists a non-constant function f ∈ L^{p}(R^{n}) such that ℳ (f) = f if and only if n ≥ 3 and n/(n − 2) < p ≤ ∞.

It is interesting to know when given an r.i. space X(R^{n}), the space D_{I_{2}}(X(R^{n})) is not trivial, or equivalently

(2) \overline{D_{I_{2}}(X(R^{n}))} := \{f ∈ L^{0}([0, ∞)) : ∥∫_{t}^{∞} f^{∗∗}(s)s^{2/n−1} ds∥_{\overline{X} (0,∞)} < ∞\}

is not trivial. This will be done in terms of the fundamental indices of X. We start by computing the fundamental function of D_{I_{2}}(X(R^{n})).

Lemma 3.9. Let X be an r.i. space on R^{n}, n ≥ 3. Let Y be given by (2). Then

ϕ_{Y} (s) ≃ s^{n/2}∥P_{1−2/n}χ_{[0,s]}∥_{X}

where P_{1−2/n}f(t) = t^{2/n−1} ∫_{0}^{t} f(s)s^{−2/n} ds.

Proof.

\begin{aligned}s^{n/2}P_{1−2/n}χ_{[0,s]}(t) & ≃ s^{n/2}(χ_{[0,s]}(t) + (\frac{s}{t})^{1−2/n}χ_{[s,∞)}(t)) \\ & ≃ ∫_{t}^{∞} χ_{[0,s]}^{∗∗}(r)r^{2/n−1} dr. \end{aligned}

Theorem 3.10. Let X be an r.i. space on R^{n}, n ≥ 3. Let Y be given by (2). Then

(1) If β̄_{X} < 1 − 2/n, then Y ≠ {0}.

(2) If Y ≠ \{0\} then \underline{β}_{X} ≤ 1 − 2/n.

Proof. (1) Let χ_{r} = χ_{[0,r]} . Then

P_{1−2/n}χ_{r}(t) = ∫_{0}^{1} χ_{r}(ξt)\frac{dξ}{ξ^{n/2}} ≤ c ∑_{k=0}^{∞} 2^{−k(1−n/2)}χ_{2^{k}r}(t).

Thus

∥P_{1−2/n}χ_{r}∥_{X} ≤ c ∑_{k=0}^{∞} 2^{−k(1−n/2)}ϕ_{X}(2^{k}r) ≤ cϕ_{X}(r) ∑_{k=0}^{∞} 2^{−k(1−n/2)}M_{X}(2^{k}).

Let ε > 0 be such that β̄_{X} + ε < 1 − 2/n. Then by the definition of β̄_{X} it follows readily that there is a constant c > 0 such that

M_{X}(2^{k}) ≤ c2^{k(β̄_{X}+ε)},

and hence

∑_{k=0}^{∞} 2^{−k(1−n/2)}M_{X}(2^{k}) ≤ ∑_{k=0}^{∞} 2^{−k(1−n/2−β̄_{X}−ε)} < ∞,

which implies that χ_{r} ∈ Y .

(2) Since Y ≠ {0} if and only if ∥P_{1−2/n}χ_{[0,1]}∥_{X} < ∞ and

(3)  sup_{t>0}(P_{1−2/n}χ_{[0,1]})^{∗∗}(t)ϕ_{X}(t) ≤ ∥P_{1−2/n}χ_{[0,1]}∥_{X} < ∞,

Invariant spaces with fixed points 45

and easy computations show that (3) implies that

(4)  1 ≤ sup_{t≥1} \frac{ϕ_{X}(t)}{t^{1−2/n}} = c < ∞,

then, by (4)

M_{X}(a) = max( sup_{t≥1/a} \frac{ϕ_{X}(ta)}{ϕ_{X}(t)}, sup_{t<1/a} \frac{ϕ_{X}(ta)}{ϕ_{X}(t)})

= max( sup_{t≥1/a} \frac{ϕ_{X}(ta)}{(at)^{1−2/n}} \frac{(at)^{1−2/n}}{ϕ_{X}(t)} , sup_{t<1/a} \frac{ϕ_{X}(ta)}{ϕ_{X}(t)})

≃ max(a^{1−2/n} sup_{t≥1/a} \frac{t^{1−2/n}}{ϕ_{X}(t)} , sup_{t<1/a} \frac{ϕ_{X}(ta)}{ϕ_{X}(t)}).

Thus, if a < 1, using again (4) we get

M_{X}(a) ≥ a^{1−2/n} sup_{t≥1/a} \frac{t^{1−2/n}}{ϕ_{X}(t)} ≥ a^{1−2/n}

which implies that

\underline{β}_{X} ≤ 1 − 2/n.

Let us see that the converse in the previous theorem is not true.

Proposition 3.11. There are rearrangement invariant spaces X such that (1) Y ≠ {0} and β̄_{X} ≥ 1 − 2/n.

(2) Y = \{0\} and \underline{β}_{X} < 1 − 2/n.

Proof. Let φ(t) = t^{a}χ_{[0,1]}(t) + t^{b}χ_{[1,∞)}(t), with 0 ≤ a, b ≤ 1. Let

X = {f ∈ L^{0}([0, ∞)) : sup_{t>0} f^{∗∗}(t)φ(t) < ∞}.

Since φ is a quasi-concave function, we have that

φ(t) = ϕ_{X}(t)

and

\underline{β}_{X} = min(a, b), β̄_{X} = max(a, b).

On the other hand, the space Y defined by (2) is not trivial if and only if

b ≤ 1 − 2/n. Now, to prove (1) take b ≤ 1 − 2/n and a ≥ 1 − 2/n. And to see (2) take

b > 1 − 2/n and a ≤ 1 − 2/n.

46 Joaquim Martín and Javier Soria

Remark 3.12. If we consider

X_{0} = {f ∈ L^{0}([0, ∞)) : sup_{t>0} f^{∗∗}(t)t^{1−2/n}(1 + log^{+} t) < ∞}

and

X_{1} = \{f ∈ L^{0}([0, ∞)) : sup_{t>0} f^{∗∗}(t)\frac{t^{1−2/n}}{(1 + log^{+} t)} < ∞\}

then \underline{β}_{X_{i}} = β̄_{X_{i}} = 1 − 2/n, Y_{0} = \{0\} and Y_{1} ≠ \{0\}.

Remark 3.13. It was proved in [7] that if we consider the strong maximal function (i.e., the maximal operator associated to centered intervals in R^{n}), then there were no fixed points in any L^{p}(R^{n}) space, regardless of the dimension. The same argument works to show that L^{p}(R^{n}) cannot be replaced by any different r.i. space. Also, if we study this question for other kind of sets, like, e.g., Buseman– Feller differentiation bases (see [6]), then the only possible fixed points are the constant functions. This observation applies to any non-centered maximal operator (with respect to balls, cubes, etc.).

References

[1] Bennett, C., and K. Rudnick: On Lorentz–Zygmund Spaces. - Dissertationes Math. (Rozprawy Mat.) 175, 1980.

[2] Bennett, C., and R. Sharpley: Interpolation of Operators. - Academic Press, Boston, 1988.

[3] Carro, M., L. Pick, J. Soria, and V. Stepanov: On embeddings between classical Lorentz spaces. - Math. Inequal. Appl. 4, 2001, 397–428.

[4] Edmunds, D. E., R. Kerman, and L. Pick: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. - J. Funct. Anal. 170, 2000, 307–355.

[5] García-Cuerva, J., and J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. - North-Holland Math. Stud. 116, North-Holland, Amsterdam, 1985.

[6] de Guzmán, M.: Differentiation of Integrals in R^{n}. - Lecture Notes in Math. 481, Springer-Verlag, Berlin–New York, 1975.

[7] Korry, S.: Fixed points of the Hardy–Littlewood maximal operator. - Collect. Math. 52, 2001, 289–294.

[8] Lieb, E. H., and M. Loss: Analysis. - American Mathematical Society, Providence, 2001.

[9] Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. - Studia Math. 96, 1990, 145–158.

[10] Stein, E.: Singular Integrals and Differentiability Properties of Functions. - Princeton University Press, New Jersey, 1970.

Received 9 September 2004